Office of Naval Research

Contract Nonr-220(57)

NR-064-487

Technical Report No. 8

TRANSIENT EXCITATION OF AN ELASTIC HALF-SPACE

BY A POINT LOAD TRAVELING ON THE SURFACE

by

D. C. Gakenheimer and J. Miklowitz

Distribution of this Document is Unlimited.

Division of Engineering and Applied Science

CALIFORNIA INSTITUTE OF TECHNOLOGY

Pasadena, California

Jo nuary 1969

Office of Naval Research

Contract Nonr-220(57)

NR-064-487

Technical Report No. 8

Transient Excitation of an Elastic Half-Space

by a Point Load Traveling on the Surfac e

by

D. C. Gakenheimer and J. Miklowitz

Reproduction in whole or in part is permitted for any purpose of the United States Government.

Distribution of this document is unlimited.

Division of Engineering and Applied Science California Institute of Technology

Pasadena, California

January 1969

ABSTRACT

The propagation of transient waves in a homogeneous, isotropic,

linearly elastic half-space excited by a traveling normal point load is

investigated. The load is suddenly applied and then it moves rectilinearly

at a constant speed along the free surface. The displacements are de­

rived for the interior of the half-space and for all load speeds. Wave

front expansions are obtained from the exact solution, in addition to

results pertaining to the steady-state displacement field. The limit

case of zero load speed is considered, yielding new results for Lamb's

point load problem.

1. INTRODUCTION

Ground motions excited by moving surface forces arise, for

example, from nuclear blasts and from shock waves generated by super­

sonic aircraft, and they interact with structures causing extensive

damage. A mathematical problem of fundamental importance in these

applications is that of an elastic half-space whose surface is excited

by a normal point load which is suddenly applied and which subsequently

moves rectilinearly at a constant speed.

In recent years several solutions to this problem have been given

for the surface of the half-space. First Payton ( 1]t computed the

transient surface displacements by using an elastodynamic reciprocal

theorem. Then Lansing ( 2] rederived some of Payton's results by

employing a Duhamel superposition integral. However, no transient

solutions have been given for the interior of the half-space, which are

t Numbers in brackets designate References at end of paper.

-2-

of interest with regard to buried structures. Therefore, it was appropriate

to seek the interior displacements here. The remaining contributions to

this problem are the steady-state results given by Mandel and Avramesco

[ 3] , Papadopoulos [ 4] , Grimes [ 5] , Eason [ 6] , and Lansing [ 2].

Of further interest is the fact that a point load moving on the sur­

face of a half-space generates a non-axisymmetric disturbance . Very few

wave propagation problems of this type have been solved and no general

solution techniques are available. Beyond the moving load cases cited

above, mention is made of Chao's work [ 7] on the surface displacements

due to a tangential surface point load. However, like the moving load

cases, Chao's solution technique is restricted to the surface of the half­

space, besides involving a separation of variables which is peculiar to

his excitation. Further, Scott and Miklowitz [ 8] have given a general

method for solving non-axisymmetric wave propagation problems involving

plates, but their technique was found not to be appropriate for the prob­

lem considered here. On the other hand, the solution technique developed

here is applicable to all points of the half-space and it is sufficiently

general that it should contribute guidelines for analyzing other non-axi­

symmetric half-space problems.

After formulating the problem in Section 2, a formal solution

is obtained in Section 3 by using the Laplace and double Fourier trans­

forms. Then in Section 4 the inverse transforms are evaluated by a

technique due originally to Cagniard [ 9] , but simplified by a transfor­

mation introduced by DeHoop [ 10] for problems in acoustics and later

used by Mitra [ 11] for an elastic half-space problem. In this way each

displacement for the interior of the half- space is reduced to a sum of

single integrals and algebraic terms for all values of the load speed. Each

contribution to the displacements is identified as a specific wave.

-3-

In particular, the integrals represent waves which emanate from the

initial position of the load as if they were generated by a stationary

point source, while the algebraic terms represent disturbances that

trail behind the load and whose wave geometry depends on the speed

of the load relative to the body wave speeds. In Section 5 this form

of the solution is exploited to evaluate the displacements near the

wave fronts. Then in Section 6 the limit case of zero load speed is

considered, showing that the integrals become a solution of Lamb's

point load problem. Finally, in Section 7 the algebraic terms are

shown to form the steady- state displacement field when the load speed

exceeds both of the body wave speeds.

2. FORMULATION OF THE PROBLEM

The subject half-space problem is depicted in Fig. 1 based on a

cartesian coordinate system (x,y,z). The plane surface of the half-

space is z = 0, with z > 0 forming the interior. A concentrated,

normal load of unit magnitude travels on the surface along the positive

x-axis at a constant speed c. The load acquires its velocity instan-

taneously at the origin of the coordinates at time t = O.

The half-space is a homogeneous, isotropic medium governed

by the equations of the linear theory of elasticity. The equations of

motion for the case of vanishing body forces can be taken as the wave

equations

2 2 1 a lJJ

Y' ljJ = - --­c2 at2

s

(1)

(characters underscored by a tilde designate vectors), where cp and l)J,

-4-

, known as the Lame potentials, are related to the displacement vector u

by

(2)

and satisfy the divergence condition

(3)

The constants cd and cs, defined by c! = (X. + 2µ)/v and c; = µ/v,

represent the dilatational and equivoluminal body wave speeds, respec­

tively, where X. and µ are the Lam~ constants and v is the material

density. The stresses T .. are related to the displacements by lJ

T •• = X.u. ko .. +µ(u . . +u .. ) lJ .K , lJ l , J J , l

where tensor notation is employed.

The boundary conditions at z = 0 take the form

T (x,y,O,t) = -o(y)o(x- ct) zz

T (x,y,O,t) = T (x,y,O,t) = 0 xz yz . l

(4)

( 5)

where o is the Dirac delta function. To represent quiescence at t = 0,

the initial conditions appear as

"'' O) _ 8cf>(x,y,z,O) 't'\x ' y' z ' - at = ljJ(x , y , z , 0) =

a~(x,y,z,O)

at = 0 • ( 6)

Finally, the potentials cf> and ljJ, and the space derivatives of the paten-

tials, are required to vanish at infinity.

3. FORMAL SOL UT ION

A solution of the wave equations (1) that satisfies the initial

conditions (6) and the boundedness condition at infinity can be computed

-5-

by using the Laplace and double Fourier transforms to suppress the

time parameter and the x,y space coordinates. Then satisfying the

boundary conditions (5), using the displacement-potential relation (2),

and inverting the Fourier transform give the Laplace transformed

displacements as

for j =x,y,z, where

1 Soos -n z+i(kx+vy) u. (x,y,z,p) = 2

F.,,,(k,v,p)e a dk dv JO' (2ir) µ J ....

for a = d, s, in which

Fxd(k,v,p) =

F yd(k,v,p) =

Fzd(k,v,p) =

-oo

-ikn G F (k,v,p) = 0 XS

-ivn G F (k,v,p) = 0 ys

ndn0

G F (k,v,p) = ZS

1 G= (p+ick)T

k = ...£._ s c s

2ikndnsG

2ivndnsG

2 2 -2nd(k +v )G ,

(7)

( 8)

(9)

(10)

(11)

(12)

( 13)

u. and u. are Laplace transform pairs, p is the Laplace transform J J

parameter, k and v are the Fourier transform parameters, and the

square roots nd and ns are assigned the branch that has the positive,

-6-

real part. In this form u. is expressed as the sum of a dilatational J

contribution ujd and an equivoluminal contribution u .• JS

4. INVERSION

The Laplace transform is inverted by a technique due to Cagniard

[ 9] , but modified for the present application. This technique consists

of converting each u. into the Laplace transform of a known function, JO'

and then inverting the Laplace transform by inspection. In the subse-

quent calculations p is assumed to be a real, positive number. For

such values of p, Lerch's theorem (see Carslaw and Jaeger [ 12])

guarantees that if u. exists, it is unique. JO!

and

To simplify the form of u. , the transformations JO!

f3 = q c 0 s a - w sin a er = q s in a + w c 0 s a

are substituted successively into (8), yielding

where

-L(m z-iqr) 1sc:orco Cd a U:. (r,9,z,p) =-2 J K.,,,(q,w,e)e dqdw,

JO! 0 -oo J ...

= -( iqcos 9(iqcos 9 +y) +w2sin2e] m L 0

K (q, w, 9) = 2[ iq cos 9(iq cos 9 +y) + w 2sin2

e] mdm L XS S

K d(q,w,9) = -sin0[ iq(iqcos 9 + -y) -w2cos 9] m L y 0

K (q, w, 9) = 2 sin a[ iq(iq cos a +y) - w 2cos a] mdm L ys s

(14)

(15)

( 16)

(1 7a)

(1 7b)

(1 7c)

(1 7d)

-7-

K (q,w,9) = -2(iqcos 9 +-y)(q2

+w2)mdL

ZS

L= 2 2 2 2 'TT cµ[(iq cos 9 +-y) + w sin 9] R

1

(1 7e)

(1 7f)

(18)

(19)

2 2 .!_ md = (q +w +1) 2

, m s

m 0

2 2 2] = ( 1 +Z(q +w ) , (20)

c s

(21)

and (r, 9, z) are the cylindrical coordinates shown in Fig. 1. The trans-

formation in (15) was introduced by DeHoop [ 10] in order to simplify

Cagniard 's technique.

In view of the symmetry properties

u (r,9,z,p) =u (r,-9,z,p) xa xa

u (r,9,z,p) = -u (r,-9, z ,p) ya ya

(22)

u (r,9,z,p) =u (r,-9,z,p), za za

u. , and hence u., are only inverted for 0 :S 9 <'TT. Since u. has different Ja J Ja

forms depending on the speed of the load relative to the body wave speeds,

the inversion of each u. is separated into three cases. In particular, Ja

the terms supersonic, transonic, and subsonic refer to the cases when

the load speed is greater than the dilatational wave speed (c > cd),

between the dilatational and equivoluminal wave speeds (cd > c > cs),

and less than the equivoluminal wave speed (c < c ) , respectively. In s

-8-

the remainder of this section uzd and u ZS

are inverted for the interior

of the half-space (z > 0) and for all load speeds (0 .:S c < oo). Then uzd

and u are combined to give uz, and similar results are displayed ZS

for u and u • x y

Dilatational Contribution for Supersonic Load Motion

From (16)

S(X)s(X) _ _E...c (mdz-iqr) - 1 d uzd(~,p) = 2

0 -oo Kzd(q,w, S)e dq dw , (23)

where ~ is the position vector. uzd is converted into the Laplace

transform of a known function by mapping (1 /cd)(mdz-iqr) into t

through a contour integration in a complex q-plane. To this end, the

singularities of the integrand of U:zd are branch points at

± q = Q , and simple poles s

± ± at q = Qc and q = QR, where

± 2 .! Qd = ±i(w +1) 2

Q±_±wsin0+iy c - cos e

± q = Q and

d

( 24)

± The poles at q = QR correspond to the zeros of the Rayleigh function R,

where YR = cd/cR and cR is the Rayleigh surface wave speed. The

roots of these singularities which lie in the upper-half of the q-plane

are shown in Fig. 2.

By seeking a particular contour in the q-plane such that

t = -1-(m z - iqr)

Cd d

one finds, upon solving for q, that

(25)

(26)

-9-

for t > twd, where

Equation (26) defines one branch of a hyperbola with vertex

2 .!. q = i(w + 1) 2 r/p and asymptotes arg q = ±r/z. As shown in Fig. 2

(27)

+ -by a solid line labeled with qd and qd, this hyperbola is parametrically

described by t as t varies from twd towards infinity. Since

r/p < 1, the hyperbola does not intersect the branch cuts in the q-plane.

The arcs c1

and CII are introduced as shown in Fig. 2 to form a

+ -closed contour C, where C = Re q-axis + CI + qd + qd + CII9

(1)

(2)

(3)

± The poles at q = Q lie inside C if, and only if, c

y cos e

(or x > O)

2 (or t > tL' where t = L) L ex

zed 2 2 i w tan 8 > - 2- (t - t ) z wd

p

(28)

For fixed t and z > 0, t = tL defines the surface of a hemisphere with

center (x = ct/2, n = 0) and radius ct/2 . Conditions (2) and (3 ) are

. 1 th d. . 2 > 2 h 2 ( 2 2 2) 2 28/ 2 2 equiva ent to e con itl.on w w od, w ere w od= p '{ -x z cos x n

2 2 .!. and n = (y + z ) 2 • To incorporate these conditions into the contour

integration for uzd. the half-space is separated into three regions:

c Region I: x > 0, ~ > --2

p c

-10-

± The poles at q = Q lie inside C for w€ [ O,oo).

c

c Region II: x > 0, ~ < --2

p c

± The poles at q = Q lie inside C for w € (w d' oo) c 0

andtheylieoutside C for w€[O,w0d).

Region III: x < 0

No poles lie inside C for w ~ [ 0, oo).

(28)

For x > 0, the rays x/p = cd/c form the surface of a cone whose axis

is the positive x-axis. This conical surface is shown in Fig. 3a along

with the part of t = tL which is bounded by x/p < cd/c, and Roman

numerals which depict the location of these regions in the half-space

(each Roman numeral lies between the rays that define the corresponding

region). The next step is to invert "i:izd for each of these regions.

Region I. The Cauchy-Goursat theorem and residue theory

applied to the integrand of uzd and c yield

u d(x,p) =A d(x,p) +13 d(x,p) z - z - z -

where

and

in which

(30)

(31)

-11-

(33)

Azd is the contribution from ± +

qd, where qd = qd, and Bzd is the

± poles at q = Q • The integrals that

c residue contribution from the

arise along CI and CII vanish as these contours recede to infinity.

By interchanging the order of integration in (31) and inverting

the Laplace transform, one finds

dw (34)

where

T = (~ -d t2

d

(3 5)

and H is the Heaviside function. Azd is a hemispherical, dilatational

wave in that it represents the disturbance behind the wave front at t = td,

where td is the arrival time of a hemispherical, dilatational wave. This

wave emanates from the initial position of the load as shownt in Fig. 3a.

The inversion of Bzd is also done by a Cagniard technique. The

singularities in the integrand of B zd are shown in Fig. 4 with branch

points at W : s: and W = s: I and simple poles at W = s~ J where

± sd = -i-y sin e ± i(1

s± = 1

-iy sin 8 ± i(1 2 - -y 2)zcos e s

s± -R- - i )' sin e ± 2 2 .!.

i()' --y )' R

cos e

± In Fig. 4 the convention is adopted that Sa and

the same function, but whose positions are w =

± g represent

O'

± S for c > c

O' O'

(36)

roots of

and

w = s: for c <ca, where a= d,s,R. Here, since c > cd, only the s:

t Arrival times are designated in the text by t subsripted with lower case letters and wave fronts are shown in the figures by solid lines.

-12-

± arise and the ga are given later in the text.

± The poles at w = SR

+ correspond to the roots of the Rayleigh function R in which q = Q • c

The particular contour in the w-plane is given by

or solving for w

t = ;d (mdz - iqr) I + q=Q c

w = w~ = -i-y sine + y c~s e (i~y ± zad) n

for t > tdc, where

[

2 - l 2 ( · zj2 = ~ - c2 - 1)n Cd

~ = ct - x

(3 7)

(38)

(39)

Equation (38) defines one branch of a hyperbola with vertex l

w = -i-y sin 9 + i(y/n)(1 - -y2)2 cos 9 and asymptotes arg w = ±yjz. In

view of the limits of integration in (32), only the plus root of (38) is

+ needed and w d = w d. As shown in Fig. 4 by the solid contour labeled

for region I, the wd part of the hyperbola is parametrically described

by t as t varies from tdc towards infinity. Since x/p > cd/c and

y/n < 1, wd intersects the imaginary w-axis below the branch point

at w = S~ and above the real w-axis. As shown in Fig. 3a, tdc

represents the arrival time of a conical, dilatational wave which trails

behind the load.

The contours C0

and Cr are introduced as shown in Fig. 4 to

-13-

form a closed contour which includes w d and the real w-axis. Then

the application of the Cauchy-Goursat theorem to the integrand of Bzd

and this closed contour produces, upon inverting the Laplace transform,

(40)

where

/\ /\ + K d(w,0) = K d(Q ,w,0) z z c

( 41)

The integral that arises along Cr vanishes as Cr recedes to infinity

and the one along C vanishes because its real part is zero. 0

Finally,

Bzd represents a conical, dilatational wave trailing behind the load.

The sum of Azd and Bzd gives uzd for region I as

Td dq -

uzd(~,t) = H(t-ta)JO Re[ Kzd(qd,w,0) dtd jaw

r /\ dwdl + Re L Kzd(w d, 0) at H(t - tdc) (42)

Region II. By noting the conditions for region II in (29) and com-

pleting the contour integration in the q-plane, one finds

Azd(~,p) =

and

Bzd(~,p) = -..E....(m z-iqr)

Res: rnzd(q,w,B)e Cd d J I dw

oJ- + q=O c

(43)

(44)

where uzd is the sum of Azd and Bzd" The letter P precedes an

improper integral to imply that it is interpreted in the sense of a Cauchy

principal value. Such an integral arises in Azd because the poles at

-14-

± ± q = Qc coalesce on qd at t = tL as w - w od" Beyond this, Azd

is exactly the same as for region I, while Bzd only differs in the lower

limit, which is expected since the poles at q

for w > w0d.

= Q± only lie inside c

c

The inversion of Bzd proceeds as for region I, but with modi­

fications in the geometry of the w-plane. For region II the particular

contour w d intersects the real w-axis as shown in Fig. 4, to comply

with the limits of integration in Bzd. Furthermore, the singularities

+ + + Sd, Ss, and SR may lie below the real w-axis, but still on the imaginary

w-axis, in a manner which is not shown in Fig. 4. However, since the

particular contour does not intersect the imaginary w-axis, this has no

bearing on the contour integration for region II. Then inverting Azd

and Bzd, and combining the results give uzd for region II as

dw

( 45)

The first term in uzd represents a hemispherical, dilatational wave

as for region I. However, for region II the integral is interpreted as a

Cauchy principal value for t = tL. The second term has the same

algebraic form as for region I, but the conical wave front is replaced w ith

the hemispherical surface t = tL. This new surface is not expected to be a

wave front (i.e., uzd and all the derivatives of uzd are expected to

be continuous through t = tL) because it is not a characteristic surface

(similar to t = td), or the closure of characteristic surfaces (similar

to t = tdc), associated with the governing wave equation for the dilatational

potential cf> in (1).

-15-

Region III. As indicated in (29) for region Ill no poles lie inside

C. Therefore, the inversion of uzd proceeds exactly as for region I,

less the residue term Bzd' and uzd can be obtained from (42) by

deleting the algebraic term.

Summary. By comparing the results for each region it follows

that uzd can be represented by one expression for all three regions;

namely,

(46)

This expression is valid for 0 < 0 < ir and z > O. The wave pattern

associated with uzd is shown in Fig. 3a, where the relationship

between the rays x/ p ::: c d/c, the wave fronts t ::: td and t ::: tdc, the

hemispherical surface t = tL, and the regions I-III is depicted.

Dilatational Contribution for Transonic and Subsonic Load Motion

The inversion of u d for c < c < cd and c < c , or equivalently z s s

c < cd, proceeds exactly as for c > cd, except that of the three regions

in (29) only II and III are applicable for c < c d because x/ p is always

less than cd/c. These two regions are depicted in Fig. 3b by Roman

numerals two and three.

The inversion of uzd for region II proceeds for c < cd as it

did for c > c d, except that the geometry of the w-plane is different.

As shown in Fig. 4, the position of the singularities varies depending

on the value of c relative to cs, cd, and cR. In that figure

-16-

± 2 l

g :::: - i l' sin 9 ± ( )' - 1) 2 cos 9 d

g: 9 ± ( l' 2 2 .!.

8 :::: - i 'Y sin - .P. ) 2 cos (47)

± ( 'Y

2 2 .!. g :::: - i )' sin 8 ± - -yR) z cos 8

R

However, the particular contour in thew-plane for region II remains the

same for all load speeds (c >cd and c <cd), as does the result of a contou r

integration which includes it and the real w-axis. Therefore, uzd for

region II and c < cd is the same as given for c > cd in (45). Furthermore,

the inversion of uzd for region III is independent of the value of the load

± speed (which only appears through the position of the poles at q :::: Q ) c

and the result is the same as given for c > c d. Then combining the

results for regions II and III gives uzd for c < cd as

(48)

where 0 .5 8 <TT and z > 0. The dilatational wave pattern for c < c d is

shown in Fig. 3b. As expected physically, the conical wave front does

not exist, leaving only the hemispherical one.

Finally, by comparing (46) and (48) one finds that uzd is given

by (46) for all load speeds if the Heaviside function satisfies the

condition

(49)

-17-

Equivoluminal Contribution

The inversion of uzs proceeds as for uzd, but uzs is more

complicated than uzd due to the appearance of head waves (or von Schmit

waves).

With u being given by (16) for j = z and a = s, the corre­zs

spending q-plane is shown in Fig. 2, where the singularities are the

same as for uzd' but the particular contour has two possible configura­

tions, whicharedenotedbydashedlines. If r/p<cs/cd and w<C._(O,oo)

or if r/p > cs/cd and w<C._ (w 1 ,oo), where w 1 = (r2

12

- p2)i/z, then

± ± the particular contour is similar to qd and it is denoted by qs, where

± Cd 2 2 l q = -

2 [itr ± z(t - t ) Z]

s ws

for t > t , in which ws

p

t ws

However, if r /p > cs/cd and w <C._ (0, w 1), then the vertex of

+

( 50)

( 5 1)

± l" q ies s

+ on the branch cut between the branch points at q = Qd and q = Q , and

s ±

the particular contour is given by qs plus qsd, where

for t > t > t d, in which ws ws

As shown in Fig. 2, the closed contour C now includes

it arises , and the real q-axis. As for uzd, the poles at q =

( 52)

( 53 )

qsd w hen

Q± can lie c

-18-

either inside or outside of C.

± In regard to the various positions of the poles at q = Qc and the

vertex of q±, the inversion ofu is separated into seven cases, each S ZS

of which corresponds to a particular region of the half- space:

Region I: x > 0, c

~>~ p c

c ~>~ r c

The poles lie inside C for wrc__ [ 0 ,oo).

The vertex does not lie on the branch cut for wrc__ [ 0, oo).

c Region II: x > 0, ~ > 2

p c

c ~>~ r c

The poles lie inside C for wrc__ [ 0, oo).

The vertex lies on the branch cut for w re__ [O,w 1).

c Region III: x > 0, ~ > 2 p c •

c ~<~ r c

The vertex lies on the branch cut for wrc__ [O,w1).

The poles lie inside C for wrc__ (0, oo) and coalesce on qsd as

w - O. The poles also coalesce on qsd as 9 - 0 for

2 l. cs < c < c d and w re__ [ 0 , ( '{ - 1) 2 ) •

Region IV: x > 0, c

~<2 p c

The poles lie inside C for wrc__ (w , oo). OS

The vertex does not lie on the branch cut for wrc__ [ 0, oo).

Region V: x > 0, c

~<2 p c

The poles lie inside C for w re__ (w , oo). OS

The vertex lies on the branch cut for wrc__ [ O,w1).

2 1 The poles coalesce on qsdas e-o for c<cdand wrc__{w

0s,('t -1) 2

).

-19-

Region VI: x < 0,

No poles lie inside C for w E:: [O, co).

The vertex does not lie on the branch cut for wrc._ [O,co).

Region VII: x < 0 1

No poles lie inside C for wrc._ [O,co).

The vertex lies on the branch cut for w re._ (0, w 1).

(Regions I - VII are referred to in the text by (54) )

2 L. 2 2 .!. In ( 54), w = (p 'Y -xi. ) 2 z cos e/xn. For x > 0, the rays x/p = c /c

OS S

form the surface of a cone which is similar to x/p = c d/c, and x /r = cd/c

defines two planes, one for 0 > 0 and one for 0 < O. Also, r/p = c s /cd

defines the surface of a cone whos e axis is the positive z-axis. All seven

regions arise when inverting u for c > cd' but only regions III-VII and ZS

regions IV-VII arise for cs< c < cd and c <cs' respectively. The

Roman numerals in Figs. 5 - 8 depict the location of these regions in

the half-space.

By inverting u ZS

for e ach region, accounting for the variabl e

load speed, and combining the results, one finds

+ H(t-t )H(~ - cs\ H (c d - ~) - H(t-t ) H(~ - cs ) -jH(x) (5 5) E p cl c r sc p c

-20-

for 0 < e s 1T. z > 0 • and 0 < c < 00. Further details on inverting

are given in a thesis by the first author [ 13] • The symbols in

which have not been previously defined are

/\ K (w,9)=

ZS - 2 sec a m ( 2 + w 2) I ircµR d q

+ q=Q c

w s

. . e + y cos e c· t + ) = - i y s in - 2 i.,, y z as n

Wsd= -i)' sin 9 + iyc~S 9 (sy- ZO'Sd) n

A = sd

1

) z -: 2

- 1 n J ,

0 for t < t s

T for t > t s s

l

; c (t-t ) 2 -t T = i (d sd+1)-1j

sd 1_ r

l 1

u ZS

u ZS

(56)

(57)

( 58)

(59)

(60)

t SC

1[ 2 2 2 2 2 -t = - ( ~ -~) z +(~ -1) y + x J sdc c 2 2 2 '

CS Cd Cd

(c~ 1

- 1)2

<j>c Cd

(62) =

(c~ 1

- 1)2 c

s

( 6 1)

-21-

The Heaviside function in (55) is restricted by the conditions

H(t - t ) = 1 if c < c SC S

(63)

H( Y.. - cf> ) = 1 if c < Cd • n c

The wave geometry associated with u is shown in Figs. 5 - 9. ZS

The dilatational wave fronts are included in these diagrams for reference;

therefore, they display the wave geometry associated with the total dis-

placement u • z However, the Roman numerals in these diagrams only

correspond to u and the regions described in (54). ZS

The first and third terms in u , which are analogous to the ZS

dilatational waves in uzd, represent hemispherical and conical, equi-

voluminal waves, respectively. The second term in u is the contri-zs

bution from qsd along the imaginary axis in the q-plane and it represents

a head wave whose wave front is the surface of the truncated cone given

by t = tsd for r/p > cs/cd. This wave, referred to as the conical head

wave, is generated by the surface intersection of the dilatational wave

front at t = td and it propagates in front of t = tB, as shown in Figs. 7

and 8(a,b); thereby, contributing both ahead and behind the equivoluminal

wave front at t = t s

surface of a sphere with center (r l

= 0,

For fixed time,

2 l_ z = cdt/2(£ -1) 2

)

t = tB is the

and radius

cdt/2(12-1)

2• The surface t = tB is not expected to be a wave front

because it is not a characteristic surface, or the closure of characteristic

surfaces, of the wave equation for the equivoluminal potential '1! in (1).

The integral in the second term of u is improper for t = t ZS S

because its integrand contains a first-order singularity at w = 0 (i.e.,

its integrand behaves like 1/w as w - O). This singularity is intro-

duced by the differential of qsd• In Section 5 this integral is evaluated

-22-

for t - t • displaying a logaritlunic singularity in u for t = t • This S ZS S

integral is also improper for certain points in the plane under the path

of the load (the 8 = 0 plane). In particular, it is interpreted as a

Cauchyprincipalvaluefor t <t<t0

d if 8=0, x/p>c le, and SC S C s'

cs < c < cd; and for tL < t < t~dc if 8 = 0, x/p < cs/c, and c <ed.

where t0

d equals t d evaluated for e = 0 and t = t 0d is shown s c s c s c

in Fig. 8(a, b) by a dashed line projecting out from the load. The

± improper integral for 8 = 0 reflects the way the poles at q = Q c

coalesce on qsd as 8 - 0, and as described in (54) for regions III and

V. Furthermore, if the Cauchy principal value is evaluated for 8 - 0,

it combines with the remaining terms in

as a - 0, which is expected physically.

u ZS

to render u ZS

continuous

The integral in the second term of u is also improper for ZS

t = tE if c > cs and x > 0 because its integrand has a first-order

singularity at w = 0 (note tE < ts, therefore Asd = 0). This singularity

± reflects the way the poles at q = Q coalesce on q d as w - 0, and as

c s

described in (54) for region III. The significance of t = tE is discussed

in connection with the last term in u • ZS

The last term in u represents another head wave. This term ZS

is the contribution from the contour w sd which is indicated in Fig. 4

as a segment of the imaginary w-axis. The dashed contour that is shown

in this figure along with w sd is the particular contour which arises in the

w-plane for u and region II if y/n >cf> (the rays y/n = <f> are ZS C C

shown in Fig. 9). This particular contour has other configurations for

region II if y /n < cf> (in this case the w d segment does not arise), c s

and for the other regions in (54), but they are not displayed here.·

This second head wave has a plane surface for a wave front

which is given by t = t d for y/n > <f> and x/r > cd/c. This wave, s c c

-23-

referred to as the plane head wave, only exists for c > c d and then

it is generated by the surface intersection of the dilatational wave

front at t = tdc

For x/r = cd/c

and it propagates in front of t = t , as shown in Fig. 9. SC

the plane head wave front is tangent to the conical he ad

wave front and for x/r < cd/c it does not exist because its generator,

t = tdc, does not exist, as shown in Fig. 5. In the region bounded by

x/r < cd/c and x/p > cs/c the plane head wave ends. This is re­

flected in u by the fact that the last term is discontinuous for t = t ZS E

and the integral in the second term is improper for t = tE. The sur-

face defined by t = tE, which is not shown in any of the figures, is not

expected to be a wave front for the same reason as given above for

For cs < c < cd the plane head wave front does not exist,

which can be seen in (55) since now H((x/r) - (cd/c)) = O. Physically

this is expected since its generator, t = tdc, does not exist. In this

case the last term in u still represents an equivoluminal distur­zs

bance propagating in front of t = t • However, now it propagates SC

behind the surface t = tE' which is behind the conical head wave front

at t = tsd, and which makes the last term indistinguishable as a separate

wave. Finally, for c < c the last term does not contribute to u , S ZS

which can be seen in (55) since H((x/r) - (cd/c)) = 0 and H((x/p) - (c8/c)) = O.

Total Displacement Field

The total disturbance comprising u z is the sum of uzd and

u ZS

Similar expressions can be computed for u and u , and all three x y

displacements are written as

6 u.(x,t) = \ u.~(x,t)

J - 6 Jt-' -13=1

-24-

(j = x' y' z) (64)

for 0 :$ r < 00, 0 :$ 9 < TT (or -OO < X < 00, 0 < y < 00), 0 < Z < 00, and

0 :S c < oo, where

(65)

(66)

(68)

[

dw -u. 5 (x,t) = Re .Q. (w , 9) dt s j H(t-t )H(t-tL)H(x)

J - JS S SC (69)

c c c -+ H(t-t )H(~ - 2-)H(~-~ )- H(t-t )H(~ -2) J H(x) • (70) E p c c r sc p c

The symbols which have not been previously defined are

= y sec am I TTcµR o

+ q = Q c

A K (w,S)=-2ysec9 j

xs trcµR mdms +

q = Q c

A 2 K d ( w , 9) = sec :. ( y sin 9 - i w) m

y TTCµ 0

q = Q+ c

(71)

(72)

(73)

(67)

/\ K (w,9) =

ys -2 sec

2e ircµR

-25-

( '{ sin e - (74)

In this form the first three terms in u. represent a system of waves J

which emanate from the initial position of the load as if they were

generated by a stationary point source. On the other hand, the last

three terms represent disturbances that trail behind the load and whose

wave geometry depends on the speed of the load relative to the body

wave speeds.

Finally, it is noted that the interior displacements given here

are not uniformly valid in z, as z - 0, because the particular con-

+ tours in the q- and w-planes wrap around the poles at q = QR and

w = s; as z - O. Therefore, to obtain the surface displacements it

is necessary to return to the formal solution in (7) and {8), set z = 0,

and then invert the displacements {or equivalently, evaluate the contri­

bution of the poles at q = o; and w = s; as z - 0, and combine

the results with the interior displacements in (64) - {70) evaluated for

z = 0). However, since Payton [ 1] and Lansing [ 2] have already

investigated the surface displacements, they are not displayed he re;

but they are given in [ 13] in context with the technique used here.

5. WAVE FRONT EXPANSIONS

The wave form of the solution given in {64) - (70) is advantageous

for evaluating the displacements near the wave fronts. To facilitate

these expansions for the waves emanating from the initial position of

the load, the time dependence in the limits of the integrals in (65) - {70)

is removed by the transformation

-26-

2 2 2 . 2 .!. w = ( A + (B - A ) s m QI] 2 (75)

where A is the lower limit and B is the upper limit of the particular

integral in question, and the QI range of integration is 0 to rr/2. Further-

more, this transformation removes a half-order singularity in the inte-

grand of uji and uj 2 which is introduced by the differentials of qd

and qs. Then uji' uj 2 , and uj3 are evaluated as t - td, t- ts, and

t - tsd, respectively, by first expanding the appropriate integrands and

then integrating these expansions from 0 to rr/2. Special attention must

be given to uj3 since the integral in this term is improper for t = t .

s

This term is evaluated by first approximating the improper integral for

t near t and then letting t - t • In this way one finds s s

uj1 = Aj1 + O(t - td) as t - t d (76)

c -rj 2 +0(t-\) for .E <~ p Cd

uj2 - as t - t s

O~t-ts)!) c

A' + for .E > ~ J2 p Cd

(77)

Aj3(t-tsd) + o((t-tsd)2) c

uj3 = as t - t for .E > ~ sd p Cd

(78)

log (If-_ 1 I) c

uj3 = AJ3 + 0(1) as t - t for .E > ~ s p Cd s

(79)

for j = x, y, z, where the wave front coefficients are given in the appendix.

These expansions are valid for 0 < 9 < rr, z > 0, and 0 < c < oo, except

that (7 6) is not valid along the rays x/p = cd/c since A . . is un­Jl

bounded along these rays. Similarly, (77) and (79) are not valid along

x/p = cs/c, and (78) is not valid along x/r = cd/c.

As the expansions in (76) - (78) show, u. has a step discontinuity J

-27-

across the hemispherical wave fronts at t = td and t = ts, and it is

continuous through t = tsd" However, for r /p > cs/cd the disturbance

near t = t is uominated by the logarithmic singularity shown in (79). s

This singularity is two-sided in that u. becomes unbounded as t-t J s

for t < t and for t '> t , and it is symmetric about t = t . s s s

Since the waves trailing behind the load are in an algebraic form,

they are expanded algebraically near their wave fronts, yielding

1

as t - tdc

for Y > <f> n c

(80)

as t - t , (81) SC

u. 6 = A!6

(t - t)--z + 0(1) J J SC

as t - t for Y > <f> SC n C

(83)

for j = x,y,z. These expansions are valid for 0 < 8 < ir and z > 0,

except that (80) only arises for c > cd and x/p > cd/c, the upper one

in (81) for c > cd and x/p > cs/c, the lower one in (81) and (83) for

c > cs and x/p > cs/c, and (82) for c > cd and x/r > cd/c. These

latter conditions correspond to when and where the wave fronts t = tdc,

t = t , and t = t d arise in the half-space. SC S C

As (80) and (81) show, u. has a half-order singularity behind J

the conical wave fronts at t = tdc and t = tsc In addition, u. con­J

tains a half-order singularity in front of t = t , as (83) shows, pro­se

ducing another two-sided singularity in uj" If c > cd, this two-sided

singularity exists across t = t for y / n > <f> ; if cs < c < c d , it SC C

exists across the entire conical wave front at t = t · and if c < cs, sc'

-28-

it does not exist at all. Also, u. is continuous through the head wave J

front at t = tsdc, as it is through t = tsd" Finally, it should be noted

that the disturbance near the wave fronts trailing behind the load is a

half-order stronger than near the corresponding (compare wave fronts

involving the same type of disturbance) wave fronts emanating from the

initial position of the load. This can be seen, for example, by com-

paring the wave front expansions of uji as t - td and uj4 as t- tdc"

6. STATIONARY POINT LOAD

A special case of the moving load problem is that of a half-space

whose surface is excited by a stationary point load with step time depen-

dence. A solution to this problem, which was first worked on by Lamb

[ 14] , is obtained here by letting c - 0 in (64) - (70) and expressing the

displacements in their cylindrical components, to give

3

u<T(~,t) =I uo-rl~·t) {3= 1

(o- = r,z)

for O<r<oo, 0:50:5tr, and 0< z<oo, where

dw

T

uo- 2 (~,t) = H(t-ts)S s Re[K~s(qs,w) :~s J dw

0

in which

(84)

(85)

(86)

(87)

-iqm M , 0

mdm0

M ,

-29-

1 M =----2

ir c dµR

l (89)

(90)

The wave geometry associated with these displacements is the same as

for c < c , as shown in Figs. 6b and Sb, except that the contours s

0 t = tL and t = tsdc are not present in the c = 0 solution.

It should be noted that Eason [ 15] , and Cinelli and Fugelso [ 16]

have also worked on Lamb's point load problem for the interior of the

half-space using transforms. Their results are also in the form of

single integrals, but they do not readily display the w ave fronts associ-

ated with a concentrated surface load. Mitra [ 11] has derived the

interior displacements for a half-space whose surface is excited by a

uniform disk of pressure. He used Cagniard's technique and DeHoop's

transformation, but his solution technique is adapted to the axisym-

metry of his problem. Furthermore, he mentions that his solution

technique is applicable to Lamb's point load problem, but he gives no

results. Also, Pekeris [ 17] has thoroughly analyzed the surface dis-

placements due to a stationary point load. His results agree in detail

with those given here if the latter are extended to include z = O.

Finally, Lang [ 18,19] and Craggs [ 20] have also worked on Lamb's

problem• but they did not use transforms.

Wave front expansions which correspond to a stationary point

load can be computed from the integrals in (86) - (88), or they can be

obtained from the expansions in Section 5. In fact, these expansions

-30-

have the same form as those given for uj 1 , uj2

, and uj3 in {76) - {79);

the only difference is that the coefficients Aj 1 , Aj 2 , Aj 2 , Aj3 , and Aj 3

must be evaluated for c = O. The wave front expansions for a stationary

point load were also computed by Knopoff and Gilbert [ 21]. They used

a Tauberian theorem and the saddle point technique to evaluate a formal

transform solution near the wave fronts. Their results agree in detail

with those that are obtained here, except that they did not detect the

logarithmic singularity at t = t for r/p > c /cd, as first noted by s s

Aggarwal and Ablow [ 22].

7. STEADY-STATE RESPONSE

By algebraic manipulations it can be shown that uj4 , ujS, and

uj6

in (68) - {70), less H(x) and H(t - tL), are only functions of the

coordinates ~, y, z which are invariant to the translation of the load

(see [ 13] for details). Therefore, these terms are constant at a fixed

position in a coordinate system moving with the load. For c > cd and

long time {t - oo), H(t = tL) = 1 and H{x) = 1 for points near the

load, and the load "runs away" from the waves emanating from the

initial position of the load. Consequently, for c > cd the waves trailing

behind the load represent the steady-state displacement field {i.e. ,

uj = uj 4 + ujS + uj 6 as t - oo for c > cd and points near the position of

the load, j = x,y,z). The wave geometry corresponding to these

steady-state displacements is shown in Fig. 9, where only the waves

that trail behind the load are shown.

Lansing [ 3] has also computed the steady-state displacements

due to a moving point load. He used a technique which assumed steady-

stateness from the outset and his results are in the form of single

-31-

integrals. As a special case, in the plane under the path of the load

(y = 0 plane), and for :\. = µ and c > c d, Lansing integrated his r e sults,

leaving the displacements in an algebraic form which agrees with that

one obtained here.

Finally, for c < cd the waves trailing behind the load do not

represent the entire steady-state displacement field. In this case the

waves emanating from the initial position of the load also contribute

and they must be evaluated for t - oo.

-32-

REFERENCES

1. Payton, R. G. , "An Application of the Dynamic Betti-Rayleigh Reciprocal Theorem to Moving-Point Loads in Elastic Media, 11

Quart. Appl. Math., Vol. 21, 1964, pp. 299-313.

2. Lansing, D. L., "The Displacements in an Elastic Half-Space Due to a Moving Concentrated Normal Load," NASA Technical Report, NASA TR R-238, 1966.

3. Mandel, J. and Avramesco, A., "Deplacements Produits par une Charge Mobile a la Surface d 'un Semi-Espace Elastique, II Compt. Rend. Acad. Sci., Paris, Vol. 252, pt. 3, 1961, pp. 3 7 3 0 - 3 7 3 5 •

4. Papadopoulos, M. , "The Elastodynamics of Moving Loads," J. Austral. Math. Soc., Vol. 3, 1963, pp. 79-92.

5. Grimes, C. K., "Studies on the Propagation of Elastic Waves in Solid Media," Ph.D. Thesis, California Institute of Technology, Pasadena, California, 1964.

6. Eason, G., "The Stresses Produced in a Semi-Infinite Solid by a Moving Surface Force," Int. J. Engng. Sci. , Vol. 2, 1965, pp. 581-609.

7. Chao, C. C. , "Dynamic Response of an Elastic Half-Space to Tangential Surface Loadings," J. Appl. Mech., Vol. 27, 1960, pp. 559-567.

8. Scott, R. A. and Miklowitz, J., "Transient Non-Axisymmetric Wave Propagation in an Infinite Isotropic Elastic Plate," in press, Int. J. Solids and Structures.

9. Cagniard, L., Reflection and Refraction of Progressive Seismic Waves, Translated by E. A. Flinn and C. H. Dix, McGraw ­Hill Book Co. , New York, 1962.

10. DeHoop, A. T., "A Modification of Cagniard's Method for Solving Seismic Pulse Problems, 11 Appl. Sci. Res. , Section B, Vol. 8, 1959, pp. 349-356.

11. Mitra, M. , "Disturbance Produced in an Elastic Half- Space by Impulsive Normal Pressure, 11 Proc. Camb. Phil. Soc., Vol. 60, 1964, pp. 683-696.

12. Car slaw, H. S. and Jaeger, J. C., Operational Methods in Applied Mathematics, Oxford University Press, 1941.

13. Gakenheimer, D. C., "Transient Excitation of an Elastic Half-Space By a Point Load Traveling on the Surface," Ph.D. Thesis, California Institute of Technology, Pasadena, California, 1968.

-33-

14. Lamb, H., "On the Propagation of Tremors Over the Surface of an Elastic Solid," Phil. Trans. Roy. Soc. Lond. (A), Vol. 203, 1904, pp. 1-42.

15. Eason, G., "The Displacements Produced in an Elastic Half­Space by a Suddenly Applied Surface Force,• J. Inst. Maths Applies, Vol. 2, 1966, pp. 299-326.

16. Cinelli, G. and Fugelso, L. E. , "Theoretical Study of Ground Motion Produced by Nuclear Blasts, 11 Mech. Res. Div. American Machine and Foundry Company, Report under Contract AF 29(601)-1190 with AF SWC, 1959.

17. Pekeris, C. L., "The Seismic Surface Pulse," Proc. Nat. Acad. Sci., Vol. 14, 1955, pp. 469-480.

18. Lang, H. A., "The Complete Solution for an Elastic Half-Space Under a Point Step Load," Rep. No. P-1141, Rand Corp. , Santa Monica, California, 1957.

19. Lang, H. A., "Surface Displacements in an Elastic Half-Space," ZAMM, Vol. 41, 1961, pp. 141-153.

20. Craggs, J. W., "On Axially Symmetric Waves, III. Elastic Waves in a Half-Space," Proc. Cambridge Philos. Soc., Vol. 59, 1963, pp. 803-809.

21. Knopoff, L. and Gilbert, F. , "First Motion Methods in Theoretical Seismology," J. Acoust. Soc. Amer. Vol. 31, 1959, pp. 1161-1168.

22. Aggarwal, H. R. and Ablow, C. M., "Solution to a Class of Three-Dimensional Pulse Propagation Problems in an Elastic Half-Space," Int. J. Engng Sci., Vol. 5, 1967, pp. 663-679.

-34-

APPENDIX

Wave Front Coefficients

The wave front coefficients are written in one expression for all

three displacements like the components of a vector.

(91)

(92)

(93)

(94)

(95)

(96)

(99)

(100)

in which

1

( n 2 2 r2)z-a2 = x p -

a -3 -

2 2 2 b 3 = {y Md + n )

2 2 2 2 h

4 = [ n (M - 1) - 2y M ] s s

M -d -

-35-

2 2 a

5 = (p - Zr )

b ( 2M2 + n2) 6 = y s

1

b8 = (yMsd- zMd)2

e - (1. 2 - 2) 2 -

( 101)

( 10 2)

( 103)

(104)

-36-

y

z

FIGURE 1

Traveling Point Load Problem

--- c

/ x /

e Oc

Im q

branch cut

Q~

. Q~ t = tws \

......... ..._ -......___ -q; I I

qsd t f qsd

"'-t=twsd q; '...__

I I + Qd

_.,... -

FIGURE 2

.,..,..... ,,.,-'

- q+ s

Contour Integration in the q- Plane

/

+ e Oc

I VJ -.J I

Re q

x

III

z = 0 Plane

x

z = 0 Plane

y

t = t d

-38-

t - t - d

(a )

y

t=t d

( b)

FIGURE 3

z

t = t L

y= 0 Plane

----c t=t

de

x

-+-- c

x

y = 0 Plane

Dilatational Wave Pattern for (a) Supersonic Load Motion and

(b) Transonic and Subsonic Load Motion

Im w

------------------s+ s

t = t SC

---I =tsdc ~ ~:

d

t =tdc Co

g+ R

s;

s;

-3 9 -

__.. - -- "'s

Region I

g,+ s

g+ d

+ s~

+ s-a

FIGURE 4

t = 00

for c > c0

for c < c a

Contour Integration in the w- Plane

Rew

}a- d ,s, R

t = t L

-40-

x

FIGURE 5

Wave Pattern in the Surface Plane

for Supersonic Load Motion

y

t = t L

-41-

/ I

x

(a)

x

' "'

( b)

FIGURE 6

VII

t = t 5

VII

y

y

Wave Pattern in the Surface Plane for (a) Transonic Load Motion and

(b) Subsonic Load Motion

/ t = tsv / I I

I I I I I VI I

z \ X Cs -p=-c-

r Cs \P=°Cci

\x Cd p=c

FIGURE 7

Wave Pattern in the Plane Under the Path of the Load for

Supersonic Load Motion

__.._._c

x

t = f sc

I

~ N I

-43-

z ( a )

t =

0

t = t sdc

z (b)

FIGURE 8

---c

-....--c

Wave Pattern in the Plane Under the Path of the Load for

(a) Transonic Load Motion and (b) Subsonic Load Motion

x

x

-44-

FIGURE 9

Waves Trailing Behind the Load for

Supersonic Load Motion

c

x

-45-

ACKNOWLEDGMENT

The first author is indebted to the National Aeronautics and

Space Administration for the support of a Graduate Traineeship.